Science (English Medium) Class 11 - CBSE Question Bank Solutions

Find the standard deviation of the first n natural numbers.

The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results:
Number of observations = 25, mean = 18.2 seconds, standard deviation = 3.25 seconds. Further, another set of 15 observations x₁, x₂, ..., x₁₅, also in seconds, is now available and we have `sum_(i = 1)^15 x_i` = 279 and `sum_(i  = 1)^15 x^2` = 5524. Calculate the standard derivation based on all 40 observations.

The mean and standard deviation of a set of n₁ observations are `barx_1` and s₁, respectively while the mean and standard deviation of another set of n₂ observations are `barx_2` and  s₂, respectively. Show that the standard deviation of the combined set of (n₁ + n₂) observations is given by

S.D. = `sqrt((n_1(s_1)^2 + n_2(s_2)^2)/(n_1 + n_2) + (n_1n_2 (barx_1 - barx_2)^2)/(n_1 + n_2)^2)`

Two sets each of 20 observations, have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and standard deviation 7 hours. Find the overall standard deviation.

If for distribution `sum(x - 5)` = 3, `sum(x - 5)^2` = 43 and total number of items is 18. Find the mean and standard deviation.

Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.

The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is ______.

Let x₁, x₂, ..., x_n be n observations and `barx` be their arithmetic mean. The formula for the standard deviation is given by ______.

Let a, b, c, d, e be the observations with mean m and standard deviation s. The standard deviation of the observations a + k, b + k, c + k, d + k, e + k is ______.

Let x₁, x₂, x₃, x₄, x₅ be the observations with mean m and standard deviation s. The standard deviation of the observations kx₁, kx₂, kx₃, kx₄, kx₅ is ______.

Let x₁, x₂, ... xn be n observations. Let wi = lx_i + k for i = 1, 2, ...n, where l and k are constants. If the mean of x_i’s is 48 and their standard deviation is 12, the mean of w_i’s is 55 and standard deviation of w_i’s is 15, the values of l and k should be ______.

Standard deviations for first 10 natural numbers is ______.

Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25 respectively. Difference of their standard deviation is ______.

If the variance of a data is 121, then the standard deviation of the data is ______.

The standard deviation of a data is ______ of any change in orgin, but is ______ on the change of scale.

The standard deviation is ______to the mean deviation taken from the arithmetic mean.

The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

Evaluate the following limit.

`lim_(x -> pi) (sin(pi - x))/(pi (pi - x))`

Evaluate the following limit.

`lim_(x ->0) cos x/(pi - x)`